Chapter Introduction to Valuation: The Time Value of Money McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc All rights reserved Chapter Outline • Time and Money • Future Value and Compounding • Present Value and Discounting • More about Present and Future Values Chapter Outline • Time and Money • Future Value and Compounding • Present Value and Discounting • More about Present and Future Values Time and Money The single most important skill for a student to learn in this course is the manipulation of money through time Time and Money We will use the time line to visually represent items over time Let’s start with fruit… yes, fruit! Time and Money If I gave you apples, one per year, then you can easily conclude that I have given you a total of three apples Visually this would look like: Today Year Years Time and Money But money doesn’t work this way If I gave you $100 each year, how much would you have, in total? $300, right? Today Year Years Time and Money But money doesn’t work this way If I gave you $100 each year, how much would you have, in total? $300, right? Today Year Years Time and Money The difference between money and fruit is that money can work for you over time, earning interest Today Year Years Time and Money Which would you rather receive: A or B? A B Today Today Year Year Years Years Finding the Number of Periods  Start with the basic equation and solve for t (remember your logs) FV = PV(1 + r)t t = ln(FV / PV) / ln(1 + r)  You can use the financial keys on the calculator as well; just remember the sign convention Number of Periods: Example You want to purchase a new car, and you are willing to pay $20,000 If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? I/Y = 10; PV = -15,000; FV = 20,000 CPT N = 3.02 years Number of Periods: Example Suppose you want to buy a new house You currently have $15,000, and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year How long will it be before you have enough money for the down payment and closing costs? Number of Periods: Example (Continued) How much you need to have in the future? Down payment = 1(150,000) = 15,000 Closing costs = 05(150,000 – 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750 Compute the number of periods PV = -15,000; FV = 21,750; I/Y = 7.5 CPT N = 5.14 years Using the formula t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years Quick Quiz IV  When might you want to compute the number of periods?  Suppose you want to buy some new furniture for your family room You currently have $500, and the furniture you want costs $600 If you can earn 6%, how long will you have to wait if you don’t add any additional money? Spreadsheet Example  Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv)  The formula icon is very useful when you can’t remember the exact formula  Click on the Excel icon to open a spreadsheet containing four different examples Finance Formulas Work the Web Many financial calculators are available online Click on the web surfer to go to Investopedia’s web site and work the following example: You need $50,000 in 10 years If you can earn 6% interest, how much you need to invest today? You should get $27,919.74 Comprehensive Problem  You have $10,000 to invest for five years  How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return?  How long will it take your $10,000 to double in value if it earns 5% annually?  What annual rate has been earned if $1,000 grows into $4,000 in 20 years? Terminology Future Value Present Value Compounding Discounting Simple Interest Compound Interest Discount Rate Required Rate of Return Formulas FV = PV(1 + r)t PV = FV / (1 + r)t r = (FV / PV)1/t – t = ln(FV / PV) / ln(1 + r) Key Concepts and Skills • Compute the future value of an investment made today • Compute the present value of an investment made in the future • Compute the return on an investment and the number of time periods associated with an investment What are the most important topics of this chapter? Time changes the value of money as money can be invested Money in the future is worth more than money received today Money received in the future is worth less today What are the most important topics of this chapter? The interest rate (or discount rate) and time determine the change in value of an investment The longer money is invested, the more compounding will increase the future value Questions? [...]... years How much would you have at time 5? Today 1 $1,000 2 3 4 5 $1,276.28 ? Future Values – Example 2 The effect of compounding is small for a small number of periods, but increases as the number of periods increases (Simple interest would have a future value of $1,250, for a difference of $26.28.) Future Values - Example 3 Suppose you had a relative deposit $10 at 5.5% 200 years ago 200 years ago... rate, expressed as a decimal t = number of periods Future Values: General Formula FV = PV(1 + r)t (1 + r)t = the future value interest factor Effects of Compounding  Simple interest  Compound interest  Consider the previous example:  FV with simple interest = 1,000 + 50 + 50 = $1,100  FV with compound interest = $1,102.50  The extra $2.50 comes from the interest of 05(50) = $2.50 earned on the first... earlier money on a time line  Future Value – later money on a time line  Interest rate – “exchange rate” between earlier money and later money  Discount rate  Cost of capital  Opportunity cost of capital  Required return or required rate of return Future Values Suppose you invest $1,000 for one year at 5% per year 1 Year Today 2 Years ? What is the future value in one year? $1,000 $1,050  Interest... back to the fruit analogy, receiving money over time is like receiving different fruits over time Today 1 Year 2 Years Time and Money And you don’t mix fruits in finance! Thus every time you see money spread out over time, you must think of the money as different; you can’t just add it up! Today 1 Year 2 Years Time and Money The difference between fruit (and anything else) and money is that money changes... the period rate  Interest is entered as a percent, not a decimal N = number of periods Remember to clear the registers (CLR TVM) after each problem Using Your Financial Calculator Hewlett-Packard 12C FV = future value PV = present value i = period interest rate  Interest is entered as a percent, not a decimal n = number of periods Remember to clear the registers (“f” + “CLX”) after each problem Future...Time and Money A is better because you get all of the $300 today instead of having to wait two years A B Today Today 1 Year 1 Year 2 Years 2 Years Time and Money Receiving money one year from now, or two years from now, is different than getting all the money today ... between earlier money and later money  Discount rate  Cost of capital  Opportunity cost of capital  Required return or required rate of return Future Values Suppose you invest $1,000 for one... Today Year Years Time and Money And you don’t mix fruits in finance! Thus every time you see money spread out over time, you must think of the money as different; you can’t just add it up! Today... B Today Today Year Year Years Years Time and Money A is better because you get all of the $300 today instead of having to wait two years A B Today Today Year Year Years Years Time and Money Receiving